Complex Analysis and Dynamics Seminar

Fall 2007 Schedule

Thurston's theorem for critically finite branched coverings represents an
important development in complex dynamical systems. It shows that a critically
finite rational map is rigid and gives a necessary and sufficient condition
for a critically finite branched covering to be realizable by a rational map.
In this talk, I will review the theorem and discuss some new Thurston type
results for geometrically finite and rotation type
branched coverings, all from bounded geometry point of view.

A hyperbolic manifold M may be understood as the quotient of
hyperbolic space H by a torsion-free discrete group G of
isometries of H. In dimensions 2 and 3 (i.e., when G is
a Fuchsian or Kleinian group acting on hyperbolic 2- or 3-space), the "thin"
pieces of M are understood largely by noting that for each parabolic
fixed point p of G, there is a set with a quite simple shape
called a horoball region that is invariant under the stabilizer of p in
G. Sadly(?), in dimensions greater than 3 there need not be an
invariant horoball region, but there is always an invariant Margulis region
whose more complicated shape depends only upon the stabilizer of p in
G. An explicit description of the shape of the Margulis region is given
for a parabolic fixed point p with a rank one stabilizer in a discrete
group of isometries acting on hyperbolic 4-space. The particular shape of this
region depends on the continued fraction expansion of the rotational part of
the generator of the stabilizer of p.

For a given t > 0, we show that there exist two finite index
subgroups of PSL(2,Z) which are
(1+t)-quasisymmetrically conjugate but the conjugating
homeomorphism is not conformal. This implies that for any t > 0
there are two finite regular covers of the once-punctured modular torus
T0 (or just the modular torus) and a (1+t)-quasiconformal
mapping between them that is not homotopic to a conformal map. As an application
of the above results, we show that the closure of the orbit of the basepoint in the
Teichmuller space T(S) of the punctured solenoid S under the
action of the corresponding modular group is strictly larger than the orbit
and is necessarily uncountable. This is joint work with V. Markovic.

It has been known that orientation-preserving isometries of hyperbolic (n+2)-space
have 2x2 matrix representations using the Clifford algebra Cn of n generators.
Recall that
Isom(H2)= PSL(2,R),
Isom(H3)= PSL(2,C), and
Isom(H4)= PSL(2,Q),
where we identify real numbers R with C0,
complex numbers C with C1, and quaternions
Q with C2.
These isomorphisms suggest a similar approach to the study of hyperbolic
isometries in higher dimensions. We will carry out such a study, emphasizing the 4-dimensional
case.

For a fixed ideal triangulation, the edge lengths parameterize
the Teichmuller space of surfaces with boundary. F. Luo found a family of
new coordinates of the Teichmuller space from the derivative cosine law.
Under each of the new coordinates, the Teichmuller space is an open convex
polytope. These new coordinates can be considered as an analog of R.
Penner's simplicial coordinate of the decorated Teichmuller space of
surfaces with cusps.

This is a sequel to my October 12 talk. An n-dimensional Mobius group is said to be
quasiconformally stable if its sufficiently small deformations
in Isom+(Hn) are all
quasiconformally conjugate to it. For example, in dimension 3, a Kleinian group
corresponding to the trice-punctured sphere is quasiconformally stable.
More generally, Marden has shown that any geometrically finite
Kleinian group in dimension 3 must be quasiconformally stable. Here we present an example
of a trice-punctured sphere group in dimension 4 which is geometrically
finite but not quasiconformally stable.

A metric on a compact oriented 4-manifold with no boundary is optimal
if it minimizes the square norm of the curvature tensor, while keeping the
total volume fixed. In a series of papers, LeBrun has studied the
question of which simply connected 4-manifolds admit optimal metrics that
are not Einstein (every Einstein metric on such a manifold is optimal). One
such construction leads to the following question: "Does there exist a Kleinian
group of the second kind that is a combination theorem free product of a
cyclic group of order 2 and a cyclic group of order 3, where the Hausdorff
dimension of the limit set of this group is greater than 1?" We discuss the
deformation spaces of Kleinian groups that are combination theorem free
products of two elliptic cyclic groups, including an answer to the above
question. This is joint work with Claude LeBrun.

Dec. 7: Caroline Series (University of Warwick)
Excursions into the Thin Part of Teichmuller Space

We discuss Rafi's combinatorial conditions which allow one to detect
whether a curve gets short in some surface along a Teichmuller
geodesic. We explain how these conditions can be used to compare
Teichmuller geodesics to "lines of minima." Introduced by
Kerkchoff, these are also bi-infinite paths in the Teichmuller space determined by
the same data, namely a pair of measured laminations which fill up
the surface. We get an estimate of the distance between the two
paths, leading to the result that lines of minima are Teichmuller
quasi-geodesics. This is joint work with Young Eun Choi and Kasra Rafi.